Fundamentals of Signal Processing

Systems in the Time-Domain

A discrete-time signal
sn
is delayed by
n0
samples when we write
snn0, with
n00. Choosing n0 to be negative advances the signal along the
integers. As opposed to analog delays,
discrete-time delays can only be
integer valued. In the frequency domain, delaying a signal
corresponds to a linear phase shift of the signal's
discrete-time Fourier transform:
↔snn02fn0S2f.

We use the term shift-invariant to emphasize that delays can only have
integer values in discrete-time, while in analog signals, delays can
be arbitrarily valued.

We want to concentrate on systems that are both
linear and shift-invariant. It will be these that allow us the
full power of frequency-domain analysis and implementations.
Because we have no physical constraints in "constructing" such
systems, we need only a mathematical specification. In analog
systems, the differential equation specifies the input-output
relationship in the time-domain. The corresponding discrete-time
specification is the difference equation.

The Difference Equation

yna1yn1…apynpb0xnb1xn1…bqxnq

Here, the output signal yn is related to its past values
ynl,
l1…p, and to the current and past values of the input signal
xn.
The system's characteristics are determined by the choices for the
number of coefficients p and
q and the coefficients' values
a1…ap
and
b0b1…bq.
There is an asymmetry in the coefficients: where is
a0
? This coefficient would multiply the
yn term in the
difference equation. We have essentially divided the
equation by it, which does not change the input-output
relationship. We have thus created the convention that
a0
is always one.

As opposed to differential equations, which only provide an
implicit description of a system (we must somehow
solve the differential equation), difference equations provide an
explicit way of computing the output for any
input. We simply express the difference equation by a program that
calculates each output from the previous output values, and the
current and previous inputs.